Find the monthly payment R needed to have a sinking fund accumulate the future value A. The yearly interest rate r and the time t in years are given below. Interest is compounded monthly. \( \mathrm{A}=\$ 4,000 ; \mathrm{r}=2.5 \% ; \mathrm{t}=1 \) \( \mathrm{R}=\$ \square \) (Round up to the nearest cent.)

To find the monthly payment \(R\) required to accumulate a future value \(A\) in a sinking fund with monthly compounding, we can use the formula for the future value of a series of payments. The formula for the future value \(A\) of a sinking fund is: \[ A = R \times \frac{(1 + i)^{nt} - 1}{i} \] Here, \(i\) is the monthly interest rate, \(n\) is the number of compounding periods per year, and \(t\) is the total number of years. For this problem, we have: - \(A = 4,000\) - \(r = 2.5\%\) per year, so the monthly interest rate is \(i = \frac{2.5\%}{12} = \frac{0.025}{12} \approx 0.0020833\) - \(t = 1\) year, which gives us \(n = 12\) compounding periods. Now, substituting \(n\) and \(i\) into the formula, we have: \[ 4,000 = R \times \frac{(1 + 0.0020833)^{12} - 1}{0.0020833} \] Calculating \((1 + 0.0020833)^{12}\): \[ (1 + 0.0020833)^{12} \approx 1.025423 \] So, \[ 4,000 = R \times \frac{1.025423 - 1}{0.0020833} \] Calculating \(\frac{1.025423 - 1}{0.0020833}\): \[ \frac{0.025423}{0.0020833} \approx 12.192 \] Now we can substitute this value back into the equation: \[ 4,000 = R \times 12.192 \] To find \(R\), divide both sides by 12.192: \[ R \approx \frac{4,000}{12.192} \approx 328.42 \] Thus, rounding up to the nearest cent, the monthly payment \(R\) needed is approximately: \[ \mathrm{R} \approx \$328.42 \]